Theorem imaeq2 5122

Description: Equality theorem for image. (Contributed by NM, 14-Aug-1994.)

Assertion

Ref	Expression
imaeq2	|- ( A = B -> ( C " A ) = ( C " B ) )

Proof of Theorem imaeq2

Step	Hyp	Ref	Expression
1		reseq2 5058	. . 3 |- ( A = B -> ( C |` A ) = ( C |` B ) )
2	1	rneqd 5011	. 2 |- ( A = B -> ran ( C |` A ) = ran ( C |` B ) )
3		df-ima 4787	. 2 |- ( C " A ) = ran ( C |` A )
4		df-ima 4787	. 2 |- ( C " B ) = ran ( C |` B )
5	2, 3, 4	3eqtr4g 2296	1 |- ( A = B -> ( C " A ) = ( C " B ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   ran crn 4775   |` cres 4776   "cima 4777

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-cnv 4782  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787

This theorem is used by:  imaeq2i  5124  imaeq2d  5126  fimadmfo  5624  ssimaex  5764  ssimaexg  5765  isoselem  6026  f1opw2  6296  supp0cosupp0fn  6507  fopwdom  7136  ssenen  7152  fiintim  7238  fidcenumlemrk  7271  fidcenumlemr  7272  sbthlem2  7275  isbth  7284  ennnfonelemp1  13297  ennnfonelemnn0  13313  ctinfomlemom  13318  ctinfom  13319  tgcn  15309  iscnp4  15319  cnpnei  15320  cnima  15321  cnconst2  15334  cnrest2  15337  cnptoprest  15340  txcnp  15372  txcnmpt  15374  metcnp3  15612